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Full text of "Treatise On Analysis Vol-Ii"

3   THE MODULUS FUNCTION ON A GROUP        265

contradiction, by first showing that the sequence (Ap) may be assumed to have a limit
A' <= BO , and then using (c) above to prove that A' contains the frontier S0 of B0 ;
finally remark that S0 + JS0  B0 .)
(e)   If A is any compact subset of R", show that

An(A)<2-Vn(S(A)y

where the diameter 8(A) is relative to the Euclidean distance, and Vrt is the Lebesgue
measure of the ball Bn (*' Bieberbach's inequality")- (Use (d) and the inequality

10.   With the notation of Problem 7, for any compact subset A of Rn the upper (resp.
lower) Minkowski area of A is defined to be the number

a+(A) = lim sup (An(Vr(A)) - An(A))/r

r-*0

(resp.

or(A) = lim inf (An(Vr(A)) - An(A))/r)

r-*0

where r tends to 0 through positive values.

(a)    Give an example of a compact set A such that a+(A) = 1 and a "(A) = 0. (Take
n  2, and consider a union of finite sets of the form ck -f Afc , where the sequence
(cfc) tends to 0, and for each k the set A* is a product Ifc x Jfc, where I*, J* are finite
sequences of the form (pak)Q^p^rk, (gbj)o^q^sk. Choose the sequences (cfc), (ak), (bk),
(rk), (sk) appropriately.)

(b)    Show that

or(A) ^/

(the isoperimetric inequality). (Use the Brunn-Minkowski inequality.)

(c)   Let A be a compact convex set (Section 8.5, Problem 8). Show that, for each

compact convex set B, the function of 

is convex for 0 <  < + co, by using the Brunn-Minkowski inequality and the relations
(a + ]8)A = aA H- 0A, (a 4- ]8)B = aB 4- /?B for all real numbers a, ]8. Deduce that
a+(A) = a "(A) (use Problem 8 of Section 8.5). The number a(A) = a+(A) = a~(A)
is called the Minkoswki area of A.

(d)    If the compact convex set A is contained in R"~ *, then a(A) = An_i(A).

(e)    If 0 is an interior point of a compact convex set A <= R", then

wrf1An(A) ^ a(A) ^ nrg1Xn(A)9

where rt is the maximum of the radii of closed balls with center 0 and contained in A,
and re is the minimum of the radii of closed balls with 0 which contain A. (Use the
inequalities

11.   Let G be a compact group, h a left invariant distance on G which defines the topology
of G (12.9.1) and JK, a Haar measure on G. Show that

is a bi-invariant distance on G, equivalent to h.hat 8(A + B) ^ 8(A) + 8(B).